Set Theory Talks

Russell Miller, CUNY
Interpreting a field in its Heisenberg group

The Heisenberg group G(F) of a field F is the group of upper triangular matrices in GL_3(F), with 1's along the diagonal and 0's below it. This group is obviously interpretable (indeed definable) in the field F. Mal'cev showed that one can recover F from G(F), and indeed that there is an interpretation of F in G(F) using two parameters. Any two noncommuting elements of G(F) can serve as the parameters, but Mal'cev was unable to produce an interpretation without parameters.

After introducing the notions of a computable functor and an effective interpretation, we will present joint work showing that there is an effective interpretation of each countable field in its Heisenberg group, without parameters, uniformly in F. (That is, the same formulas give the interpretation, no matter which field F we consider.) Moreover, from the effective interpretation we will then extract a traditional interpretation without parameters, in the usual model-theoretic sense. Finally we will note that, whereas Mal'cev's result actually gives a definition of F in G(F), there is no parameter-free definition of F there.

This work is joint with Rachael Alvir, Wesley Calvert, Grant Goodman, Valentina Harizanov, Julia Knight, Andrey Morozov, Alexandra Soskova, and Rose Weisshaar.

Gunter Fuchs, CUNY
The blurry HOD hierarchy

Classically, an object is ordinal definable if it is the unique one satisfying a formula with ordinal parameters. Generalizing this concept, given a cardinal κ, I call an object <κ-blurrily ordinal definable if it belongs to an ordinal definable set with fewer than κ elements (called a <κ-blurry definition). By considering the hereditary versions of this notion, one arrives at a hierarchy of inner models, indexed by cardinals κ: the collection of all hereditarily <κ-blurrily ordinal definable sets, which I call <κ-HOD. In a ZFC-model, this hierarchy spans the entire spectrum from HOD to V.

The special cases κ=ω and κ=ω1 have been previously considered, but no systematic study of the general setting has been carried out, it seems. One main aspect of the analysis is the notion of a leap, that is, a cardinal at which a new object becomes hereditarily blurrily definable.

In this talk, I will focus on the ZFC-provable structural properties of the blurry HOD hierarchy, which turn out to be surprisingly plentiful. So for the most part, the talk will be forcing-free. Time permitting, I may hint at the result of the equiconsistency between the least leap being the successor of a singular strong limit cardinal and the existence of a measurable cardinal, for which, admittedly, forcing is used in one direction.

Russell Miller, CUNY
Interpreting a field in its Heisenberg group

The Heisenberg group G(F) of a field F is the group of upper triangular matrices in GL_3(F), with 1's along the diagonal and 0's below it. This group is obviously interpretable (indeed definable) in the field F. Mal'cev showed that one can recover F from G(F), and indeed that there is an interpretation of F in G(F) using two parameters. Any two noncommuting elements of G(F) can serve as the parameters, but Mal'cev was unable to produce an interpretation without parameters.

After introducing the notions of a computable functor and an effective interpretation, we will present joint work showing that there is an effective interpretation of each countable field in its Heisenberg group, without parameters, uniformly in F. (That is, the same formulas give the interpretation, no matter which field F we consider.) Moreover, from the effective interpretation we will then extract a traditional interpretation without parameters, in the usual model-theoretic sense. Finally we will note that, whereas Mal'cev's result actually gives a definition of F in G(F), there is no parameter-free definition of F there.

This work is joint with Rachael Alvir, Wesley Calvert, Grant Goodman, Valentina Harizanov, Julia Knight, Andrey Morozov, Alexandra Soskova, and Rose Weisshaar.

Inici del missatge reenviat:

De: Joan Bagaria <joan.bagaria@icrea.cat>

Tema: Barcelona Set Theory Seminar

Data: 3 de juny de 2022, 8:56:35 CEST

Dear All, 

Please find attached the announcement of the next session of the Barcelona Set Theory Seminar. Feel free to distribute it.

SPEAKER:   William Mance

TITLE: Descriptive complexity in number theory and dynamics

DATE: 8 June 2022

TIME: 16:00 (CEST)

PLACE: The Seminar will take place online via Zoom:

https://ub-edu.zoom.us/j/98565247347?pwd=VVh2VVRzekYwZUJiS2FKNmRJYmZXQT09

Meeting ID: 985 6524 7347

Passcode: 243408

Best regards,

Joan

P.S.: If you do not wish to receive any more announcements, please send an email to bagaria@ub.edu with the text “Unsubscribe”.

Joan Bagaria 
ICREA Research Professor 
Universitat de Barcelona
Departament de Matemàtiques i Informàtica
Gran Via de les Corts Catalanes 585
08007 Barcelona
Catalonia 

Phone: +34 93 402 1609
joan.bagaria@icrea.cat
bagaria@ub.edu

Dino Rossegger, UC Berkeley and TU Wien
The structural complexity of models of PA

The Scott rank of a countable structure is the least ordinal α such that all automorphism orbits of the structure are definable by infinitary Σα formulas. Montalbán showed that the Scott rank of a structure is a robust measure of the structural and computational complexity of a structure by showing that various different measures are equivalent. For example, a structure has Scott rank α if and only if it has a Πα+1 Scott sentence if and only if it is uniformly ΔΔ0α categorical if and only if all its automorphism orbits are Σα infinitary definable.

In this talk we present results on the Scott rank of non-standard models of Peano arithmetic. We show that non-standard models of PA have Scott rank at least ω, but, other than that, there are no limits to their complexity. Given a completion T of PA we give a reduction via bi-interpretability of the class of linear orders to the models of T. This allows us to exhibit models of T of Scott rank α for every ω≤α≤ω1. In particular, every completion of T has models of high Scott rank.

This is joint work with Antonio Montalbán.

Speaker:     Gershom Bazerman, Arista Networks.

Date and Time:     Wednesday May 4, 2022, 7:00 - 8:30 PM., on Zoom.

Title:     Classes of Closed Monoidal Functors which Admit Infinite Traversals.

Weak Indestructibility and Reflection

Assuming multiple of strong cardinals, there are lots of cardinals with small degrees of strength (i.e. κ that are κ+2-strong). We can calculate the consistency strength of these all cardinal's small degrees of strength being weakly indestructible using forcing and core model techniques in a way similar to Apter and Sargsyan's previous work. This yields some easy relations between indestructibility and Woodin cardinals, and also generalizes easily to supercompacts. I will give a proof sketches of these results.

Michał Godziszewski, University of Vienna
Modal Quantifiers, Potential Infinity, and Yablo sequences

When properly arithmetized, Yablo's paradox results in a set of formulas which (with local disquotation in the background) turns out to be consistent, but ω-inconsistent. Adding either uniform disquotation or the ω-rule results in inconsistency. Since the paradox involves an infinite sequence of sentences, one might think that it doesn't arise in finitary contexts. We study whether it does. It turns out that the issue depends on how the finitistic approach is formalized. On one of them, proposed by Marcin Mostowski, all the paradoxical sentences simply fail to hold. This happens at a price: the underlying finitistic arithmetic itself is ω-inconsistent. Finally, when studied in the context of a finitistic approach which preserves the truth of standard arithmetic, the paradox strikes back - it does so with double force, for now the inconsistency can be obtained without the use of uniform disquotation or the ω-rule. This is joint work with Rafał Urbaniak from the University of Gdańsk.

Speaker:     Alex Sorokin, Northeastern University.

Date and Time:     Wednesday April 27, 2022, 7:00 - 8:30 PM., on Zoom.

Title:     The defect of a profunctor.

Abstract: In the mid 1960s Auslander introduced a notion of the defect of a finitely presented functor on a module category. In 2021 Martsinkovsky generalized it to arbitrary additive functors. In this talk I will show how to define a defect of any enriched functor with a codomain a cosmos. Under mild assumptions, the covariant (contravariant) defect functor turns out to be a left covariant (right contravariant) adjoint to the covariant (contravariant) Yoneda embedding. Both defects can be defined for any profunctor enriched in a cosmos V. They happen to be adjoints to the embeddings of V-Cat in V-Prof. Moreover, the Isbell duals of a profunctor are completely determined by the profunctor's covariant and contravariant defects. These results are based on applications of the Tensor-Hom-Cotensor adjunctions and the (co)end calculus.

Dino Rossegger, UC Berkeley and TU Wien
The structural complexity of models of PA

The Scott rank of a countable structure is the least ordinal α such that all automorphism orbits of the structure are definable by infinitary Σα formulas. Montalbán showed that the Scott rank of a structure is a robust measure of the structural and computational complexity of a structure by showing that various different measures are equivalent. For example, a structure has Scott rank α if and only if it has a Πα+1 Scott sentence if and only if it is uniformly ΔΔ0α categorical if and only if all its automorphism orbits are Σα infinitary definable.

In this talk we present results on the Scott rank of non-standard models of Peano arithmetic. We show that non-standard models of PA have Scott rank at least ω, but, other than that, there are no limits to their complexity. Given a completion T of PA we give a reduction via bi-interpretability of the class of linear orders to the models of T. This allows us to exhibit models of T of Scott rank α for every ω≤α≤ω1. In particular, every completion of T has models of high Scott rank.

This is joint work with Antonio Montalbán.

Speaker:     Gershom Bazerman, Arista Networks.

Date and Time:     Wednesday May 4, 2022, 7:00 - 8:30 PM., on Zoom.

Title:     Classes of Closed Monoidal Functors which Admit Infinite Traversals.

Stationary logic and set theory

Stationary logic was introduced in the 1970’s. It allows the quantifier 'for almost all countable subsets s…'. Although it is undoubtedly a kind of second order logic, it is completely axiomatizable, countably compact and satisfies a kind of Downward Lowenheim-Skolem theorem. In this talk I give first a general introduction to the extension of first order logic by this 'almost all'-quantifier. As 'almost all' is interpreted as 'for a club of', the theory of this logic is entangled with properties of stationary sets. I will give some examples of this. The main reason to focus on this logic in my talk is to use it to build an inner model of set theory. I will give a general introduction to this inner model, called C(aa), or the aa-model, and sketch a proof of CH in the model. My work on the aa-model is joint work with Juliette Kennedy and Menachem Magidor.

Michał Godziszewski, University of Vienna
Modal Quantifiers, Potential Infinity, and Yablo sequences

When properly arithmetized, Yablo's paradox results in a set of formulas which (with local disquotation in the background) turns out to be consistent, but ω-inconsistent. Adding either uniform disquotation or the ω-rule results in inconsistency. Since the paradox involves an infinite sequence of sentences, one might think that it doesn't arise in finitary contexts. We study whether it does. It turns out that the issue depends on how the finitistic approach is formalized. On one of them, proposed by Marcin Mostowski, all the paradoxical sentences simply fail to hold. This happens at a price: the underlying finitistic arithmetic itself is ω-inconsistent. Finally, when studied in the context of a finitistic approach which preserves the truth of standard arithmetic, the paradox strikes back - it does so with double force, for now the inconsistency can be obtained without the use of uniform disquotation or the ω-rule. This is joint work with Rafał Urbaniak from the University of Gdańsk.

The surprising strength of reflection in second-order set theory with abundant urelements

I shall give a general introduction to urelement set theory and the role of the second-order reflection principle in second-order urelement set theory GBCU and KMU. With the abundant atom axiom, asserting that the class of urelements greatly exceeds the class of pure sets, the second-order reflection principle implies the existence of a supercompact cardinal in an interpreted model of ZFC. The proof uses a reflection characterization of supercompactness: a cardinal κ is supercompact if and only if for every second-order sentence true in some structure M (of any size) is also true in a first-order elementary substructure m≺M of size less than κ. This is joint work with Bokai Yao. http://jdh.hamkins.org/surprising-strength-of-reflection-with-abundant-urelements-cuny-set-theory-seminar-april-2022

Speaker:     Morgan Rogers, Universit`a degli Studi dell’Insubria.

Date and Time:     Wednesday March 30, 2022, 7:00 - 8:30 PM., on Zoom.

Title:     Toposes of Topological Monoid Actions.

Abstract: Anyone encountering topos theory for the first time will be familiar with the fact that the category of actions of a monoid on sets is a special case of a presheaf topos. It turns out that if we equip the monoid with a topology and consider the subcategory of continuous actions, the result is still a Grothendieck topos. It is possible to characterize such toposes in terms of their points, and along the way extract canonical representing topological monoids, the complete monoids. I'll sketch the trajectory of this story, present some positive and negative results about Morita-equivalence of topological monoids, and explain how one can extract a semi-Galois theory from this set-up.

Speaker:     Joseph Dimos.

Date and Time:     Wednesday March 23, 2022, 7:00 - 8:30 PM., on Zoom.

Title:     Introduction to Fusion Categories and Some Applications.

Abstract: Tensor categories and multi-tensor categories have strong alignment with module categories. We can use the multi-tensor categories C in conjunction with classifying tensor algebras wrt C. From here, we can illustrate some examples of tensor categories: the category Vec of k-vector spaces that gives us a fusion category. This is defined as a category Rep(G) of some finite dimensional k-representations of a group G. From here, I will walk through the correspondence of tensor categories (Etingof) and fusion categories. Throughout, I will indicate a few unitary and non-unitary cases of fusion categories. Those unitary fusion categories are those that admit a uniquely monoidal structure. For example, this draws upon [Jones 1983] for finite index and finite depth that bridges a subfactor A-bimodule B to provide a full subcategory of some category A by its module structure. I will discuss some of these components throughout and explain the Morita equivalence of fusion categories.

Speaker:     Morgan Rogers, Universit`a degli Studi dell’Insubria.

Date and Time:     Wednesday March 30, 2022, 7:00 - 8:30 PM., on Zoom.

Title:     TBA.

Models of Relevant Arithmetic

In the 1970s, the logician and philosopher Robert Meyer proposed a novel response to Goedel's Incompleteness Theorems, suggesting that perhaps the results' impact could be blunted by analyzing Peano arithmetic with a weaker deductive system. Initial successes of the program of relevant arithmetic were positive. E.g., R# (the theory of Peano arithmetic under the relevant logic R) can be shown consistent in the sense of not proving 0=1 and this can be shown through arguably finitistic methods. In this talk I will discuss the rise and fall of Meyer's program, detailing the philosophical foundations, its positive development, and the context of Harvey Friedman's negative result in 1992. I'll also suggest why the program, although not necessarily successful, is nevertheless an interesting object of study.

Also note that a great deal of context—including Meyer's two long-unpublished monographs on the topic—have recently appeared in a special issue of the Australasian Journal of Logic I co-edited with Graham Priest, which can be found at https://ojs.victoria.ac.nz/ajl/issue/view/751.

Speaker:     Joseph Dimos.

Date and Time:     Wednesday March 23, 2022, 7:00 - 8:30 PM., on Zoom.

Title:     Introduction to Fusion Categories and Some Applications.

Abstract: Tensor categories and multi-tensor categories have strong alignment with module categories. We can use the multi-tensor categories C in conjunction with classifying tensor algebras wrt C. From here, we can illustrate some examples of tensor categories: the category Vec of k-vector spaces that gives us a fusion category. This is defined as a category Rep(G) of some finite dimensional k-representations of a group G. From here, I will walk through the correspondence of tensor categories (Etingof) and fusion categories. Throughout, I will indicate a few unitary and non-unitary cases of fusion categories. Those unitary fusion categories are those that admit a uniquely monoidal structure. For example, this draws upon [Jones 1983] for finite index and finite depth that bridges a subfactor A-bimodule B to provide a full subcategory of some category A by its module structure. I will discuss some of these components throughout and explain the Morita equivalence of fusion categories.

Joel David Hamkins, Notre Dame University
Infinite wordle and the mastermind numbers

I shall introduce and consider the natural infinitary variations of Wordle, Absurdle, and Mastermind. Infinite Wordle extends the familiar finite game to infinite words and transfinite play—the code-breaker aims to discover a hidden codeword selected from a dictionary Δ⊆Σω of infinite words over a countable alphabet Σ by making a sequence of successive guesswords, receiving feedback after each guess concerning its accuracy. For any dictionary using the usual 26-letter alphabet, for example, the code-breaker can win in at most 26 guesses, and more generally in n guesses for alphabets of finite size n. Meanwhile, for some dictionaries on an infinite alphabet, infinite play is required, but the code-breaker can always win by stage ω on a countable alphabet, for any fixed dictionary. Infinite Mastermind, in contrast, is a subtler game than Wordle because only the number and not the position of correct bits is given. When duplication of colors is allowed, nevertheless, then the code-breaker can still always win by stage ω, but in the no-duplication variation, no countable number of guesses (even transfinite) is sufficient for the code-breaker to win. I therefore introduce the mastermind number, denoted mm, to be the size of the smallest winning no-duplication Mastermind guessing set, a new cardinal characteristic of the continuum, which I prove is bounded below by the additivity number add(M) of the meager ideal and bounded above by the covering number cov(M). In particular, the precise value of the mastermind number is independent of ZFC and can consistently be strictly between ℵ1 and the continuum 2ℵ0. In simplified Mastermind, where the feedback given at each stage includes only the numbers of correct and incorrect bits (omitting information about rearrangements), then the corresponding simplified mastermind number is exactly the eventually different number d(≠∗). http://jdh.hamkins.org/infinite-wordle-and-the-mastermind-numbers-cuny-logic-workshop-march-2022/

We investigate which forcing notions can be embedded into a Tree-Prikry forcing. It turns out that the answer changes drastically under different large cardinal assumptions. We will focus on the class of κ-strategically closed forcings of cardinality κ, <κ-strategically closed forcings of cardinality κ and the κ-distributive forcing notions of cardinality κ. Then we will examine distributive subforcings of the Prikry forcing of cardinality larger than κ. This is a joint work with Moti Gitik and Yair Hayut.

Joel David Hamkins, Notre Dame University
Infinite wordle and the mastermind numbers

I shall introduce and consider the natural infinitary variations of Wordle, Absurdle, and Mastermind. Infinite Wordle extends the familiar finite game to infinite words and transfinite play—the code-breaker aims to discover a hidden codeword selected from a dictionary Δ⊆Σω of infinite words over a countable alphabet Σ by making a sequence of successive guesswords, receiving feedback after each guess concerning its accuracy. For any dictionary using the usual 26-letter alphabet, for example, the code-breaker can win in at most 26 guesses, and more generally in n guesses for alphabets of finite size n. Meanwhile, for some dictionaries on an infinite alphabet, infinite play is required, but the code-breaker can always win by stage ω on a countable alphabet, for any fixed dictionary. Infinite Mastermind, in contrast, is a subtler game than Wordle because only the number and not the position of correct bits is given. When duplication of colors is allowed, nevertheless, then the code-breaker can still always win by stage ω, but in the no-duplication variation, no countable number of guesses (even transfinite) is sufficient for the code-breaker to win. I therefore introduce the mastermind number, denoted mm, to be the size of the smallest winning no-duplication Mastermind guessing set, a new cardinal characteristic of the continuum, which I prove is bounded below by the additivity number add(M) of the meager ideal and bounded above by the covering number cov(M). In particular, the precise value of the mastermind number is independent of ZFC and can consistently be strictly between ℵ1 and the continuum 2ℵ0. In simplified Mastermind, where the feedback given at each stage includes only the numbers of correct and incorrect bits (omitting information about rearrangements), then the corresponding simplified mastermind number is exactly the eventually different number d(≠∗). http://jdh.hamkins.org/infinite-wordle-and-the-mastermind-numbers-cuny-logic-workshop-march-2022/